00000244.htm

9 THE CAUCHY FORMULA 227

Then the power series (c_v(z — a)^v) is absolutely summable in P and its sum
is equal to f(z).

Using Cauchy's formula, and the fact that/(0; %) = 1 (see (9.8.4)), we
have, by induction on p — k and the assumption,

(9.9.4.1) /(*!,..., x,,z_fc+1,...,z_p)

_. ¹ f _dx f _dx ... f
— /~ .-v_D-.fe I ^axk+i I ^axk+2 7

(2ni)^p *J_yfc+1 J_yk+2 J_yp(

for \Xj - aj\ = r_s (1 <y < k) and \Zj - a_}\ < r_j (k + 1 <y </>). On the
other hand, for \z_k — %| < r_fc (1 < k^p), we can write, for |^ - a_k\ = r_k

the power series on the right hand side being normally summable in the
set F defined by \x_k — a_k\*=r_k (1 <£</>)» hy (5.5.3). By induction on
p — fc, if we write

f(x_l9 ...,x_p)dx_p

we have by the mean value theorem

n_p,...₉ _m

^rk+l ^rp

if II /(^i> •••,x_p)\\ <M on F. It follows that the power series

(^₊i.-.n>i> •••>**)^ ⁱⁿ ^the _m^zJ~^aJ ^is ^ab'

solutely summable in P; using induction on;? - k, and applying (5.3.5) and
(8.7.9), we see that the sum of that series isf(x_ly ...₉x_k9 z_k+l9 . . . , z_p). The
conclusion follows by taking k = 0. Moreover (9.9.4.2), for k = 0, proves
that, with the same assumptions and notations as in (9.9.4)

(9.9.5) lk_w..._BJ<M/rJ¹---r;*

tf !!/(*) II <M on the product of the circles \x_k-a_k\=r_k (l^k^p)
(Cauchy's inequalities).

If in (9.9.4) we take A = (7, we see that

(9.9.6) An analytic mapping ofC^p into a complex Banach space is an entire
function.



	9 THE CAUCHY FORMULA 227 o Then the power series (c_v(z — a)^v) is absolutely summable in P and its sum is equal to f(z). Using Cauchy's formula, and the fact that/(0; %) = 1 (see (9.8.4)), we have, by induction on p — k and the assumption, (9.9.4.1) /(!,..., x,,z_fc+1,...,z_p) _. ¹ f _dx* f _dx ... f — /~ .-v_D-.fe I ^axk+i I ^axk+2 7 *(2ni)^p* J_yfc+1 J_yk+2* J_yp(

	for \Xj - aj\ = r_s (1 <y < k) and \Zj - a_}\ < r_j (k + 1 <y </>). On the other hand, for \z_k — %\| < r_fc (1 < k^p), we can write, for \|^ - a_k\ = r_k



	the power series on the right hand side being normally summable in the set F defined by \x_k — a_k\=r_k* (1 <£</>)» hy (5.5.3). By induction on p — fc, if we write

	f(x_l9 ...,x_p)dx_p

	we have by the mean value theorem M

	n_p,...₉ _m ^rk+l ^rp if II /(^i> •••,x_p)\\ <M on F. It follows that the power series (^₊i.-.n>i> •••>*)^ ⁱⁿ ^the _m^zJ~^aJ* ^is ^ab' solutely summable in P; using induction on;? - k, and applying (5.3.5) and (8.7.9), we see that the sum of that series isf(x_ly ...₉x_k9 z_k+l9 . . . , z_p). The conclusion follows by taking k = 0. Moreover (9.9.4.2), for k = 0, proves that, with the same assumptions and notations as in (9.9.4) (9.9.5) lk_w..._BJ<M/rJ¹---r;* tf !!/(*) II <M on the product of the circles \x_k-a_k\=r_k (l^k^p) (Cauchy's inequalities). If in (9.9.4) we take A = (7, we see that (9.9.6) An analytic mapping ofC^p into a complex Banach space is an entire function.